Properties

Label 45.20.b.a
Level $45$
Weight $20$
Character orbit 45.b
Analytic conductor $102.968$
Analytic rank $0$
Dimension $2$
CM discriminant -15
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,20,Mod(19,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 20, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.19");
 
S:= CuspForms(chi, 20);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 45.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(102.967513450\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 647 \beta q^{2} - 1568757 q^{4} - 1953125 \beta q^{5} - 675771443 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + 647 \beta q^{2} - 1568757 q^{4} - 1953125 \beta q^{5} - 675771443 \beta q^{8} + 6318359375 q^{10} + 1363640148089 q^{16} - 434935118914 \beta q^{17} + 1011668819684 q^{19} + 3063978515625 \beta q^{20} + 5718556719212 \beta q^{23} - 19073486328125 q^{25} - 135667566768232 q^{31} + 527976317505999 \beta q^{32} + 14\!\cdots\!90 q^{34} + \cdots + 73\!\cdots\!21 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3137514 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3137514 q^{4} + 12636718750 q^{10} + 2727280296178 q^{16} + 2023337639368 q^{19} - 38146972656250 q^{25} - 271335133536464 q^{31} + 28\!\cdots\!80 q^{34}+ \cdots + 35\!\cdots\!40 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
2.23607i
2.23607i
1446.74i 0 −1.56876e6 4.36732e6i 0 0 1.51107e9i 0 6.31836e9
19.2 1446.74i 0 −1.56876e6 4.36732e6i 0 0 1.51107e9i 0 6.31836e9
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.20.b.a 2
3.b odd 2 1 inner 45.20.b.a 2
5.b even 2 1 inner 45.20.b.a 2
15.d odd 2 1 CM 45.20.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.20.b.a 2 1.a even 1 1 trivial
45.20.b.a 2 3.b odd 2 1 inner
45.20.b.a 2 5.b even 2 1 inner
45.20.b.a 2 15.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2093045 \) acting on \(S_{20}^{\mathrm{new}}(45, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2093045 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 19073486328125 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 94\!\cdots\!80 \) Copy content Toggle raw display
$19$ \( (T - 1011668819684)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16\!\cdots\!20 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T + 135667566768232)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 15\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{2} + 27\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 11\!\cdots\!18)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( (T - 13\!\cdots\!96)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 90\!\cdots\!80 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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